Based on a simple convexity lemma, we develop bounds for differ(cid:173) ent types of Bayesian prediction errors for regression with Gaussian processes. The basic bounds are formulated for a fixed training set. Simpler expressions are obtained for sampling from an input distri(cid:173) bution which equals the weight function of the covariance kernel, yielding asymptotically tight results. The results are compared with numerical experiments.